Abstract
A topology optimization framework is proposed for robust design of skeletal structures with stochastically imperfect structural members. Imperfections are modeled as uncertain members’ out-of-straightness using curved frame elements in the form of predefined functions with random magnitudes throughout the structure. The stochastic perturbation method is used for propagating the imperfection uncertainty up to the structural response level, and the expected value of performance measure or constraint is used to form the stochastic topology optimization problem. Sensitivities are derived explicitly using the adjoint method and are used in conjunction with an efficient gradient-based optimizer in search for robust optimal topologies. Topological designs for three representative examples are investigated with the proposed algorithm and the resulting topologies are compared with the deterministic designs. It is observed that the new designs primarily feature load path diversification, which is pronounced with increasing level of uncertainty, and occasionally member thickening to mitigate the impact of the uncertainty in members’ out-of-straightness on structural performance.
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References
Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393
Allaire G, Dapogny C (2014) A linearized approach to worst-case design in parametric and geometric shape optimization. Math Models Methods Appl Sci 24(11):2199–2257
Amir O, Sigmund O, Lazarov BS, Schevenels M (2012) Efficient reanalysis techniques for robust topology optimization. Comput Methods Appl Mech Eng 245–246:217–231
Asadpoure A, Tootkaboni M, Guest JK (2011) Robust topology optimization of structures with uncertainties in stiffness–application to truss structures. Comput Struct 89(11):1131–1141
Ben-Tal A, Nemirovski A (1997) Robust truss topology design via semidefinite programming. SIAM J Optim 7(4):991–1016
Bendsoe MP, Sigmund O (2004) Topology optimization: theory, methods and applications. Springer, Berlin
Cambou B (1975) Application of first-order uncertainty analysis in the finite element method in linear elasticity. In: Proceedings of 2nd international conference on applications of statistics and probability in soil and structural engineering, pp 67–87
Changizi N, Jalalpour M (2017a) Robust topology optimization of frame structures under geometric or material properties uncertainties. Struct Multidiscip Optim 51(4):1–17
Changizi N, Kaboodanian H, Jalalpour M (2017b) Stress-based topology optimization of frame structures under geometric uncertainty. Comput Methods Appl Mech Eng 315(2):121–140. https://doi.org/10.1016/j.cma.2016.10.039
Chen S, Chen W, Lee S (2010) Level set based robust shape and topology optimization under random field uncertainties. Struct Multidiscip Optim 41(4):507–524
Collins JD, Thomson WT (1969) The eigenvalue problem for structural systems with statistical properties. AIAA J 7(4):642–648
Csébfalvi A (2014) A new theoretical approach for robust truss optimization with uncertain load directions. Mech Based Des Struct Mach 42(4):442–453
Deaton JD, Grandhi RV (2014) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optim 49(1):1–38
Dunning PD, Kim HA (2013) Robust topology optimization: minimization of expected and variance of compliance. AIAA J 51(11):2656–2663
Gu X, Sun G, Li G, Mao L, Li Q (2013) A comparative study on multiobjective reliable and robust optimization for crashworthiness design of vehicle structure. Struct Multidiscip Optim 48(3):669–684
Guest J, Igusa T (2008) Structural optimization under uncertain loads and nodal locations. Comput Meth Appl Mech Eng 198(1):116–124
Guo X, Bai W, Zhang W, Gao X (2009) Confidence structural robust design and optimization under stiffness and load uncertainties. Comput Methods Appl Mech Eng 198(41-44):3378–3399
Guo X, Du J, Gao X (2011) Confidence structural robust optimization by non-linear semidefinite programming-based single-level formulation. Int J Numer Methods Eng 86(8):953–974
Guo X, Zhang W, Zhong W (2014) Doing topology optimization explicitly and geometrically a new moving morphable components based framework. J Appl Mech 81(8):081,009
Hisada T, Nakagiri S (1981) Stochastic finite element method developed for structural safety and reliability. In: Proceedings of the 3rd international conference on structural safety and reliability, pp 395–408
Jalalpour M, Igusa T, Guest JK (2011) Optimal design of trusses with geometric imperfections: accounting for global instability. Int J Solids Struct 48(21):3011–3019
Jalalpour M, Guest JK, Igusa T (2013) Reliability-based topology optimization of trusses with stochastic stiffness. Struct Saf 43 :41–49
Jalalpour M, Tootkaboni M (2016) An efficient approach to reliability-based topology optimization for continua under material uncertainty. Struct Multidiscip Optim 53(4):759–772
Jang GW, Dijk NP, Keulen F (2012) Topology optimization of mems considering etching uncertainties using the level-set method. Int J Numer Methods Eng 92(6):571–588
Jansen M, Lombaert G, Diehl M, Lazarov BS, Sigmund O, Schevenels M (2013) Robust topology optimization accounting for misplacement of material. Struct Multidiscip Optim 47(3):317–333
Jansen M, Lombaert G, Schevenels M (2015) Robust topology optimization of structures with imperfect geometry based on geometric nonlinear analysis. Comput Methods Appl Mech Eng 285:452–467. https://doi.org/10.1016/j.cma.2014.11.028, http://www.sciencedirect.com/science/article/pii/S004578251400454X
Keshavarzzadeh V, Fernandez F, Tortorelli DA (2017) Topology optimization under uncertainty via non-intrusive polynomial chaos expansion. Comput Methods Appl Mech Eng 318:120–147. https://doi.org/10.1016/j.cma.2017.01.019, http://www.sciencedirect.com/science/article/pii/S0045782516313019
Kleiber M, Hien TD (1992) The stochastic finite element method: basic perturbation technique and computer implementation. Wiley, New York
Lazarov BS, Schevenels M, Sigmund O (2012) Topology optimization with geometric uncertainties by perturbation techniques. Int J Numer Methods Eng 90(11):1321–1336
Liu WK, Belytschko T, Mani A (1986) Probabilistic finite elements for nonlinear structural dynamics. Comput Methods Appl Mech Eng 56(1):61–81
Lógó J (2007) New type of optimality criteria method in case of probabilistic loading conditions. Mech Based Des Struct Mach 35(2):147–162
Lógó J, Ghaemi M, Rad MM (2009) Optimal topologies in case of probabilistic loading: the influence of load correlation. Mech Based Des Struct Mach 37(3):327–348
Lombardi M, Haftka RT (1998) Anti-optimization technique for structural design under load uncertainties. Comput Methods Appl Mech Eng 157(1-2):19–31
Luo Z, Chen LP, Yang J, Zhang YQ, Abdel-Malek K (2006) Fuzzy tolerance multilevel approach for structural topology optimization. Comput Struct 84(3):127–140
Martínez-Frutos J, Herrero-Pérez D, Kessler M, Periago F (2016) Robust shape optimization of continuous structures via the level set method. Comput Methods Appl Mech Eng 305:271–291
Medina JC, Taflanidis A (2015) Probabilistic measures for assessing appropriateness of robust design optimization solutions. Struct Multidiscip Optim 51(4):813–834
Moens D, Vandepitte D (2005) A survey of non-probabilistic uncertainty treatment in finite element analysis. Comput Methods Appl Mech Eng 194(1216):1527–1555, special Issue on Computational Methods in Stochastic Mechanics and Reliability Analysis. https://doi.org/10.1016/j.cma.2004.03.019
Richardson JN, Coelho RF, Adriaenssens S (2015) Robust topology optimization of truss structures with random loading and material properties: a multiobjective perspective. Comput Struct 154:41–47
Sandgren E, Cameron T (2002) Robust design optimization of structures through consideration of variation. Comput Struct 80(20):1605–1613
Schevenels M, Lazarov BS, Sigmund O (2011) Robust topology optimization accounting for spatially varying manufacturing errors. Comput Methods Appl Mech Eng 200(49):3613–3627
Shinozuka M, Astill CJ (1972) Random eigenvalue problems in structural analysis. AIAA J 10(4):456–462
Sigmund O, Jensen JS (2003) Systematic design of phononic band–gap materials and structures by topology optimization. Philos Trans Royal Soc Lond A: Math Phys Eng Sci 361(1806):1001–1019
Sigmund O (2009) Manufacturing tolerant topology optimization. Acta Mech Sinica 25(2):227–239
Sigmund O (2011) On the usefulness of non-gradient approaches in topology optimization. Struct Multidiscip Optim 43(5):589–596
Suzuki K, Kikuchi N (1991) A homogenization method for shape and topology optimization. Comput Methods Appl Mech Eng 93(3):291–318
The MathWorks Inc (2017) MATLAB - Optimization Toolbox, Version 7.6. The MathWorks Inc., Natick, Massachusetts, http://www.mathworks.com/products/optimization/
Tootkaboni M, Asadpoure A, Guest JK (2012) Topology optimization of continuum structures under uncertainty–a polynomial chaos approach. Comput Methods Appl Mech Eng 201:263–275
Venini P, Pingaro M (2017) An innovative h-norm based worst case scenario approach for dynamic compliance optimization with applications to viscoelastic beams. Struct Multidiscip Optim 55(5):1685–1710
Wu J, Aage N, Westermann R, Sigmund O (2018) Infill optimization for additive manufacturingapproaching bone-like porous structures. IEEE Trans Vis Comput Graph 24(2):1127–1140
Zhang W, Yuan J, Zhang J, Guo X (2016) A new topology optimization approach based on moving morphable components (mmc) and the ersatz material model. Struct Multidiscip Optim 53(6):1243–1260
Acknowledgments
This work was supported by the National Science Foundation under Grant No. CMMI-1401575. This support is gratefully acknowledged. Asadpoure also acknowledges support from University of Massachusetts College of Engineering.
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Ahmadi, B., Jalalpour, M., Asadpoure, A. et al. Robust topology optimization of skeletal structures with imperfect structural members. Struct Multidisc Optim 58, 2533–2544 (2018). https://doi.org/10.1007/s00158-018-2035-y
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DOI: https://doi.org/10.1007/s00158-018-2035-y